7/28/2019 Discrete 121206

1/20

Discrete Math

Week 15 December 6, 2012

Shimizu

7/28/2019 Discrete 121206

2/20

7/28/2019 Discrete 121206

3/20

Theorems

The Handshake Theorem: If G is any graph, thenthe sum of the degrees of all of the vertices of G

equals twice the number of edges of G.

In other words, If the vertices of G are (integer 1, 2, 3, ), then

deg 2(# )= .

Corollary to the Handshake Thm: The total degreeof a graph is even

7/28/2019 Discrete 121206

4/20

Practice

Is it possible to draw a graph with the specifiedproperties? If so, draw an example. If not, explain

why.

4 vertices of degrees 1, 1, 2, and 3. 4 vertices of degrees 1, 1, 3, and 3.

Simple graph, 4 vertices of degree 1, 1, 3, and 3.

7/28/2019 Discrete 121206

5/20

Proposition 10.1.3. In any graph there are an evennumber of vertices of odd degree.

Proof:

(start of ) Suppose G is any graph and G has n vertices of odd

degree and m vertices of even degree (n and m are

positive integers).

What do we have to show?

What else do we know about G?

7/28/2019 Discrete 121206

6/20

Definitions

Let G be a graph, and let v and w be vertices in G.

A walk from v to w is a finite alternating sequence ofadjacent vertices and edges of G.

A walk is of the form0 .

The vs represent vertices, the es representedges, 0 , , and the endpoints of

are and (for all 1,2,3, , )

A trivial walk from v to v consists of the single vertexv.

7/28/2019 Discrete 121206

7/20

Definitions

A trail from v to w : a walk that does not contain arepeated edge.

A path from v to w: a trail that does not contain a

repeated vertex.

A closed walk: a walk that starts and ends at thesame vertex.

A circuit: a closed walk that contains at least one

edge and does not contain a repeated edge.

A simple circuit: a circuit that does not have any

other repeated vertex except the first and last vertex.

7/28/2019 Discrete 121206

8/20

More definitions

Let G be a graph. A graph H is a connectedcomponent of a graph G iff

1. H is a subgraph of G;

2. H is connected; and

3. No connected subgraph of G has H as a subgraph andcontains vertices or edges that are not in H.

7/28/2019 Discrete 121206

9/20

More definitions

Let G be a graph. Two vertices v and w areconnected iff there is a walk from v to w.

The graph G is connected iff given any two vertices

v and w in G, there is a walk from v to w.

Practice: Is this graph connected?

7/28/2019 Discrete 121206

10/20

7 Bridges of Knigsberg

Is it possible for a person to take a walk around thetown of Knigsberg, start and end at the same

location and cross each of the 7 bridges exactly

once?

7/28/2019 Discrete 121206

11/20

More definitions

Let G be a graph. An Euler circuit for G is a circuitthat contains every vertex and every edge of G.

Is there an Euler circuit for the Konigsberg bridges

trail?

7/28/2019 Discrete 121206

12/20

7 Bridges of Knigsberg

Is it possible for a person to take a walk around thetown of Knigsberg, start and end at the same

location and cross each of the 7 bridges exactly

once?

7/28/2019 Discrete 121206

13/20

More definitions

Let G be a graph. A Hamiltonian circuit for G is asimple circuit that includes every vertex of G.

Is there a Hamiltonian circuit for the Konigsberg

bridges trail?

7/28/2019 Discrete 121206

14/20

Theorem 10.2.2

If a graph has an Euler circuit, then every vertex ofthe graph has a positive even degree.

Contrapositive of Thm 10.2.2:

If some vertex of a graph has odd degree, then the

graph does not have an Euler circuit.

7/28/2019 Discrete 121206

15/20

Theorem 10.2.3

If a graph G is connected and the degree of everyvertex of G is a positive even integer, then G has an

Euler circuit.

Does this graph have an Euler circuit?

7/28/2019 Discrete 121206

16/20

Theorem 10.2.4

A graph G has an Euler circuit iff G is connected andevery vertex of G has positive even degree.

7/28/2019 Discrete 121206

17/20

Homework

10.2/ (1 20)odd

Due next Wednesday, December 12.

7/28/2019 Discrete 121206

18/20

Color Square Game

C1 C2 C3 C4

R1

R2

R3

R4

7/28/2019 Discrete 121206

19/20

Chapter 10: Graphs and Trees

Definitions A graph G consists of 2 finite sets:

1. a non-empty set V(G) ofvert ices

2. a set E(G) ofedges. Each edge is associated with a set

consisting of one or two vertices, called its endpoints.

The correspondence from edges to endpoints is called the

edge-endpoint function.

7/28/2019 Discrete 121206

20/20

This is where we stand

Week # Mon. Wed. Chapters we are supposed to cover

according to the department syllabus

10 Class

cancelle

d

Class

cancelle

d

11 11/5 11/7 6.1 6.2 6.3

12 11/12 11/14 7.1 7.2 7.3

13

Thanks-

giving

week

11/19

mid-term

11/21 8.1 8.2 8.3

14 11/26 11/29 10.1 10.2

15 12/3 12/5 This means we have to do 1 to 2 sections

per class.

16 12/10 12/12 This means we will have homework over

Thanksgiving break.

17 12/17